Is tautology for logic what theorems are for mathematics?

Question

Consider the following statements. "If x is an integer then 3+2=5" and "If x is not integer then 3+2=5". Constructing truth tables for the above statements show that there is no case P is true and Q is false. So both statements are true.

Also the statement where we have substitute the above statements "If (If P then Q and If not P then Q) then Q" is true. Can we say that the whole implication or Q is a tautology? I would say no because it depends on what we have defined as "integer", "3+2=5" etc. But from the above statements we can concluded that Q is true regardless of P i.e. always true. Does it make it a tautology? Can someone help me how to distinguish them?

From what I understand (might be wrong) tautologies are about truth tables irrespective of the meaning of the statements whereas theorems are based on the meaning of the statements. Another example is the statement "If x is positive then x squared is also positivie". It is true because we can eliminate from the truth table the line with (T and F) i.e. always true but not a tautology. But also when we "search" for tautologies we search for always true statements. Do they have in common the fact that both theorems (given the set of axioms in a system) and tautologies (given the set of axioms of laws of logic) are statements that are always true?

Answer 1

Considering the most simple case of mathematical theorems and logical tautologies:

1. Theorems in mathematics are analytical true statements. They have the form “If A holds, then also B holds”.

   Example: “If two triangles have in common one side and the same two adjoining angles (A), then the triangles are congruent (B).”

   To prove a mathematical theorem means to discover from the definition of the concepts in A the property B. Therefore the proof is analytical.

2. A tautology of propositional logic is a logical formula F(A,B,…) with variables A,B,…, such that: When replacing the variables by arbitrary statements then the resulting proposition is true.

   Example: If A implies B then non-B implies non-A.

Hence both concepts, tautology and mathematical theorem, are not the same. But it is interesting to elaborate on their difference.

Answer 2

Tautology applies to propositional logic:

a formula that is always true regardless of which valuation is used for the propositional variables.

The corresponding terms for predicate logic is that of Valid formula:

a formula that is true under every possible interpretation.

According to the definitions, a tautology is a valid formula of propositional logic.

In natural language, it has little sense to say that a statement S is a tautology (in the formal sense) because it is not very useful to apply the condition of the definition: "true in every possible interpretation."

From a formal point of view, a tautology is a theorem of propositional calculus.

A valid first order formula is a theorem of predicate calculus.

For formal arithmetic, i.e. the first-order version of Peano's axioms, a formula like e.g. 2+3=5 is a theorem, because it is provable from the axioms.

The arithmetical formula 2+3=5 is not valid, because it is not true in every interpretation.

But it is a Logical consequence of the axioms of arithmetic, because it is true in every interpretation that satisfies the axioms.

Answer 3

Your statements are indeed both tautologies. Thanks to Mauro Allegranza for finding the most straightforward definition:

a formula that is always true regardless of which valuation is used for the propositional variables.

So we have a way to check if a statement is a tautology:

1. Find all the propositional variables
2. Evaluate the truth of the statement for every possible assignment of true/false among those variables.

If the statement is true for every combination of those variables, it is a tautology.

If x is an integer then 3+2=5

1. There are zero propositional variables.
2. There is one possible way to assign true/false to zero variables: the empty function.

Is the statement true after performing the requisite zero variable assignments? Yes: the statement is unchanged, and as you pointed out, it happens to be true. Therefore the statement is true for every possible assignment and is a tautology. The same holds for the second statement.

Let's consider the statement "If P then 3+2=5". Is this a tautology? Now we have non-trivial work to do in the assignment stage, because we have one propositional variable. We check P = true and P = false, and see that this (different) statement is also a tautology.

it depends on what we have defined as "integer", "3+2=5" etc

No, this is an implicit part of the statement. If we don't have fixed definitions of those terms, then we have not made a logical statement yet. The definitions of the words used is an intrinsic part of the statement. If I were to change the definition of the symbol "3", I would be making a different logical statement and we'd have to evaluate that separately.

For the sake of argument, we are all implicitly agreeing to use the standard, common definitions of those symbols. If you secretly change those definitions without telling the people you are debating/arguing/reasoning with, then you are not acting in good faith. Similarly, if you're using a non-standard definition of one of those words or symbols, you must define them when making the statement. Otherwise you risk ambiguity and a frustrating argument where the parties don't realize they're arguing about different statements.